In 1990s several researchers(Hornik, Cybenko, etc.) have proved that feedforward neural networks with bounded and non-constant activation function are able to approximate any $L_p$-Integrable function. However many popular activation functions today, ReLU for example, doesn't satisfy the bounded prerequisite. Is there any theory on approximation capability of neural networks with ReLU as activation function yet?
2026-03-28 18:16:01.1774721761
Do we have an universal approximation theory on ReLU activated Neural Networks already?
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You can simulate a node activation such as $$n(x)=\cases{0& if $x<0$\\x& if $0\leq x\leq 1$\\1& otherwise}$$ with two ReLU nodes $n_1,n_2$ like so: $$ n(x)=n_1(x)-n_2(x-1) $$ So anything that can be done with the activation $n$ can be exactly replicated with ReLU, and at most twice as many nodes.